Understanding the axioms: the interactions of the Axiom of Choice with large cardinal axioms

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Much of the research in pure mathematics is concerned with proving the existence of abstract mathematical objects from a certain set of initial assumptions, or axioms. Set theory is a branch of mathematical logic, and it serves as the mainstream foundation of mathematics. More precisely, the axioms of set theory function as a "universal interpreter" for all pure mathematical research.One of the standard axioms of set theory is the Axiom of Choice. This axiom relates to choosing an object from a collection, given that the collection is non-empty. We can easily choose an object from a single non-empty collection of objects, and inductively we can choose from any finite number of non-empty collections. However, it is not always possible to coherently describe a way to choose from infinitely many collections at once, even if we know that none of them is empty. For example, if we have finitely many pairs of ants, we can go one pair at a time and choose an ant from each pair. If we have infinitely many pairs, then there are no obvious discerning properties that let us specify, with a finite algorithm, a means of choosing a single ant from each pair. If, however, we are given infinitely many pairs, each consisting of one ant and one wasp, we can always choose the wasp.The Axiom of Choice asserts that there is always a way to make a coherent choice, but it does not provide us with a description of what this choice is. Indeed, it often doesn't even matter in proofs of existence what the exact choice of objects is. Nevertheless, we are often interested in the question of whether or not we can find a way to construct the objects whose existence we proved, since having an effective way of doing something sheds more light on the problem and its solution. This is one of the main goals in research related to the Axiom of Choice: discover the limitations of what we can and cannot construct explicitly in the mathematical universe. Despite its non-constructive nature and a history rife with controversy, its many important consequences make the Axiom of Choice a staple of modern mathematics.Another family of set-theoretic axioms is formed of the so-called "large cardinal axioms". These are axioms asserting the existence of objects - aptly referred to as "large cardinals" in most cases - which generalise the set of the natural numbers in certain kind of ways. These large cardinals are much larger than the objects mathematicians normally take interest in (such as the real numbers and so on), but their existence affects them nonetheless. There are concrete statements about natural numbers which cannot be proved without assuming that large cardinal axioms are consistent with set theory.The characterisations of large cardinals are often given from several different directions. Some are combinatorial in their nature, others are more technical. But the proofs that these characterisations are equivalent utilise the Axiom of Choice in a very significant way. We know that, in the absence of the Axiom of Choice, small cardinals may satisfy some of the combinatorial properties characterising large cardinals. And so far there has been very little research into what sort of implications there are to the existence of large cardinals when characterised by seemingly stronger properties.This project aims to explore the consequences of large cardinal axioms without the Axiom of Choice, and improve our understanding of how these axioms impact the structure of the set-theoretic universe, and the mathematical universe as a whole. Specifically, we are concerned with the question of what sort of consequences of the Axiom of Choice must follow from the existence of these large cardinals. For this we need to develop new methods that will let us explore these questions, and many others.